André plane

Let ${\displaystyle F=GF(q)}$ be a finite field, and let ${\displaystyle K=GF(q^{n})}$ be a degree ${\displaystyle n}$ extension field of ${\displaystyle F}$. Let ${\displaystyle \Gamma }$ be the group of field automorphisms of ${\displaystyle K}$ over ${\displaystyle F}$, and let ${\displaystyle \beta }$ be an arbitrary mapping from ${\displaystyle F}$ to ${\displaystyle \Gamma }$ such that ${\displaystyle \beta (1)=1}$. Finally, let ${\displaystyle N}$ be the norm function from ${\displaystyle K}$ to ${\displaystyle F}$.

Define a quasifield ${\displaystyle Q}$ with the same elements and addition as K, but with multiplication defined via ${\displaystyle a\circ b=a^{\beta (N(b))}\cdot b}$, where ${\displaystyle \cdot }$ denotes the normal field multiplication in ${\displaystyle K}$. Using this quasifield to construct a plane yields an André plane.[2]

1. André planes exist for all proper prime powers ${\displaystyle p^{n}}$ with ${\displaystyle p}$ prime and ${\displaystyle n}$ a positive integer greater than one.
2. Non-Desarguesian André planes exist for all proper prime powers except for ${\displaystyle 2^{n}}$ where ${\displaystyle n}$ is prime.

The smallest non-Desarguesian André plane has order 9, and it is isomorphic to the Hall plane of that order.

The translation planes of order 16 have all been classified, and again the only non-Desarguesian André plane is the Hall plane.[3]

There are three non-Desarguesian André planes of order 25.[4] These are the Hall plane, the regular nearfield plane, and a third plane not constructible by other techniques.[5]

There is a single André plane of order 27.[6]